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Computer Science 341 Discrete Mathematics Homework 10 Due at noon on Fri, Dec 6, 2002 No collaboration is permitted for this homework. Problem 1 Consider a group of nk politicians who must be assigned to n committees. Each politician is qualified to belong to some but not necessarily all the committees. Each committee must be assigned exactly k qualified members, and each politician must be assigned to exactly one of the committees for which that politician qualifies. Give a necessary and sufficient condition on the qualifications of the politicians for the committees for this to be possible. (Hint: For k = 1 the condition of Hall’s Theorem suffices.) Problem 2 Let X be an n-element set and let S1 , S2 , . . . , Sn be subsets of X such that |Si ∩ Sj | ≤ 1 whenever √ 1 ≤ i ≤ j ≤ n. Prove that at least one of the sets Si has size at most C n for some absolute constant C (independent of n). Problem 3 The Princeton Tigers are playing a rival team in a 2-out-of-3 series, i.e. they play games until one team wins a total of two games. The probability that the Tigers win the first game is 1/2. For subsequent games, the probability of winning depends on the outcome of the preceding game; the team is energized by victory and demoralized by defeat. If the Tigers win a game, then they have a 2/3 chance of winning the next game. On the other hand, if the team loses, they have only a 1/3 chance of winning the following game. (a) What is the probability that the Tigers win the 2-out-of-3 series given that they win the first game ? (b) What is the probability that the Tigers win the first game, given that they win the series ? Problem 4 Let S = {1, 2, . . . , n} and let π be a random permutation on S, i.e. π : S → S is a random bijection from S to S For a subset A of S, let f (A) be the minimum of π(a) over all elements a ∈ A. Let A and B be two arbitrary subsets of S. Express the following in terms of A and B. (a) Pr[f (A) = f (B)] (a) Pr[f (A) < f (B)]